Here \(t_z\)  is the tilting parameter along the \(z\) -axis. We only focus on the case when \(t^\chi _{z}<v_F\) , thus restricting ourselves to type-I Weyl semimetals.
Depending on whether the two cones are tilted along the same or opposite direction, the behavior of both LMC and PHC can be different. In the absence of strain, if the cones are tilted in opposite directions, i.e., \(t^\chi _z=-t^{\chi ^{\prime }}_z\) , a linear in magnetic field term is added to the overall longitudinal magnetoconductivity, and the parabola is shifted and tilted along a particular direction. In other words, we can say that tilting results in weak sign-reversal, although this has never been explicitly pointed out in earlier works [1]}, [2]}, [3]}. When the intervalley scattering strength is large, tilting the Weyl cones results in both weak and strong sign-reversal.
In the presence of both tilt and strain, we arrive at a very interesting scenario. Both of these parameters, i.e., \(t_z\)  and \(B_5\) , can tilt the LMC parabola either in the same direction or opposite direction, and this depends on the angle between the tilt direction and the strain induced gauge field. In Fig. REF  we plot the longitudinal magnetoconductivity for a tilted TR broken WSM presented in Eq. REF  when the Weyl cones are tilted opposite to each other. Depending on the direction of the strain induced gauge field \(B_5\) , the effects of tilting and strain can either add up or even cancel out. In Fig. REF  (a), \(B_5>0\) , and the strain and tilting effects work in opposite directions, while in Fig. REF  (b), \(B_5<0\) , and the strain and tilting effects work in the same direction.
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